Multiple Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems on Time Scales

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1 Electronic Journl of Qulittive Theory of Differentil Equtions 2009, No. 32, -3; Multiple Positive Solutions for the System of Higher Order Two-Point Boundry Vlue Problems on Time Scles K. R. Prsd, P. Murli 2 nd N.V.V.S.Surynryn 3,2 Deprtment of Applied Mthemtics Andhr University Viskhptnm, , Indi rjendr92@rediffmil.com; 2 murli uoh@yhoo.co.in 3 Deprtment of Mthemtics, VITAM College of Engineering, Viskhptnm, 5373, A.P, Indi Abstrct In this pper, we estblish the existence of t lest three positive solutions for the system of higher order boundry vlue problems on time scles by using the well-known Leggett-Willims fixed point theorem. And then, we prove the existence of t lest 2k- positive solutions for rbitrry positive integer k. Introduction The boundry vlue problems (BVPs) ply mjor role in mny fields of engineering design nd mnufcturing. Mjor estblished industries such s the utomobile, erospce, chemicl, phrmceuticl, petroleum, electronics nd communictions, s well s emerging technologies such s nnotechnology nd biotechnology rely on the BVPs to simulte complex phenomen t different scles for design nd mnufctures of high-technology products. In these pplied settings, positive solutions re meningful. Due to their importnt role in both theory nd pplictions, the BVPs hve generted gret del of interest over the recent yers. The development of the theory hs gined ttention by mny reserchers. To mention few, we list some ppers Erbe nd Wng [7], Eloe nd Henderson [5, 6], Hopkins nd Kosmtov [9], Li [0], Atici nd Guseinov [], Anderson nd Avery [2], Avery nd Peterson [3] nd Peterson, Rffoul nd Tisdell [2]. For the time scle clculus nd nottion for delt differentition, s well s Key words: Time scles, boundry vlue problem, positive solution, cone. AMS Subject Clssifiction: 39A0, 34B5, 34A40. EJQTDE, 2009 No. 32, p.

2 concepts for dynmic equtions on time scles, we refer to the introductory book on time scles by Bohner nd Peterson [4]. By n intervl we men the intersection of rel intervl with given time scle. In this pper, we ddress the question of the existence of multiple positive solutions for the nonliner system of boundry vlue problems on time scles, { y (m) + f (t, y, y 2 ) = 0, t [, b] () y2 (n) + f 2 (t, y, y 2 ) = 0, t [, b] subject to the two-point boundry conditions y (i) () = 0, 0 i m 2, y (σ q (b)) = 0, y (j) 2 () = 0, 0 j n 2, y 2 (σ q (b)) = 0, where f i : [, σ q (b)] R 2 R, i =, 2 re continuous, m, n 2, q = min{m, n}, nd σ q (b) is right dense so tht σ q (b) = σ r (b) for r q. This pper is orgnized s follows. In Section 2, we prove some lemms nd inequlities which re needed lter. In Section 3, we obtin existence nd uniqueness of solution for the BVP ()-(2), due to Schuder fixed point theorem. In Section 4, by using the cone theory techniques, we estblish sufficient conditions for the existence of t lest three positive solutions to the BVP ()-(2). The min tool in this pper is n pplictions of the Leggett-Willims fixed point theorem for opertor leving Bnch spce cone invrint, nd then, we prove the existence of t lest 2k positive solutions for rbitrry positive integer k. 2 Green s function nd bounds In this section, we construct the Green s function for the homogeneous BVP corresponding to the BVP ()-(2). And then we prove some inequlities which re needed lter. To obtin solution (y (t), y 2 (t)) of the BVP ()-(2) we need the G n (t, s), (n 2) which is the Green s function of the BVP, (2) y (n) = 0, t [, b] (3) y (i) () = 0, 0 i n 2, (4) y(σ n (b)) = 0. (5) EJQTDE, 2009 No. 32, p. 2

3 Theorem 2. The Green s function for the BVP (3)-(5) is given by G n (t, s) = (n )! n i= n i= (t σ i ())(σ n (b) σ i (s)) (σ n (b) σ i ()), t s, (t σ i ())(σ n (b) σ i (s)) (σ n (b) σ i ()) n i= (t σi (s)), σ(s) t. Proof: It is esy to check tht the BVP (3)-(5) hs only trivil solution. Let y(t, s) be the Cuchy function for y (n) = 0, nd be given by y(t, s) = (n )! σ(s) σ 2 (s)... σ n (s) } {{ } (n ) times τ τ... τ = n (t σ i (s)). (n )! For ech fixed s [, b], let u(., s) be the unique solution of the BVP Since u (n) (., s) = 0, u (i) (, s) = 0, 0 i n 2 nd u(σ n (b), s) = y(σ n (b), s). u (t) =, u 2 (t) = y(t, s) t=σ n (b)= re the solutions of u (n) = 0, u(t, s) = α (s). + α 2 (s). n (σ n (b) σ i (s)). (n )! τ,..., u n (t) = i= τ α n (s). σ()... σ n 2 () } {{ } (n ) times σ() i=... σ n 2 () } {{ } (n ) times τ τ... τ τ τ... τ By using boundry conditions, u (i) () = 0, 0 i n 2, we hve α = α 2 =... = α n = 0. Therefore, we hve n u(t, s) = α n... τ τ... τ = α n (t σ i ()). σ() σ n 2 () }{{} i= (n ) times EJQTDE, 2009 No. 32, p. 3

4 Since, it follows tht α n n u(σ n (b), s) = y(σ n (b), s), (σ n (b) σ i ()) = i= From which implies α n = Hence G n (t, s) hs the form for t s, And for t σ(s), G n (t, s) = G n (t, s) = n (n )! i= n (σ n (b) σ i (s)) (n )! (σ n (b) σ i ()). i= n (n )! i= (σ n (b) σ i (s)). (t σ i ())(σ n (b) σ i (s)). (σ n (b) σ i ()) G n (t, s) = y(t, s) + u(t, s). It follows tht n (t σ i ())(σ n (b) σ i n (s)) (t σ i (s)). (n )! (σ n (b) σ i ()) (n )! i= i= Lemm 2.2 For (t, s) [, σ n (b)] [, b], we hve Proof: For t s σ n (b), we hve Similrly, for we hve G n (t, s) = G n (t, s) G n (σ(s), s). (6) n (t σ i ())(σ n (b) σ i (s)) (n )! (σ n (b) σ i ()) i= n (n )! i= = G n (σ(s), s). (σ(s) σ i ())(σ n (b) σ i (s)) (σ n (b) σ i ()) σ(s) t σ n (b), we hve G n (t, s) G n (σ(s), s). Thus, G n (t, s) G n (σ(s), s), for ll (t, s) [, σ n (b)] [, b]. EJQTDE, 2009 No. 32, p. 4

5 Lemm 2.3 Let I = [ σn (b)+3 4, 3σn (b)+ 4 ]. For (t, s) I [, b], we hve G n (t, s) 6 n G n(σ(s), s). (7) Proof: The Green s function for the BVP (3)-(5) is given in the Theorem 2., clerly shows tht G n (t, s) > 0 on (, σ n (b)) (, b). For t s < σ n (b) nd t I, we hve n G n (t, s) G n (σ(s), s) = (t σ i ())(σ n (b) σ i (s)) (σ(s) σ i ())(σ n (b) σ i (s)) i= n i= 4 n. (t σ i ()) (σ n (b) ) And for σ(s) t < σ n (b) nd t I, we hve G n (t, s) G n (σ(s), s) = n i= (t σi ())(σ n (b) σ i (s)) n i= (t σi (s))(σ n (b) σ i ()) n i= (σ(s) σi ())(σ n (b) σ i (s)) i= (t σi ())(σ n (b) σ i (s)) n i= (t σi (s))(σ n (b) σ i ()) n i= (σn (b) σ i ())(σ n (b) σ i (s)) n [(σ(s) )(σ2 (b) t)] n i=2 (t σi ())(σ n (b) σ i (s)) n i= (σn (b) σ i ())(σ n (b) σ i ()) 6 n. Remrk: G n (t, s) G n (σ(s), s) nd G m (t, s) G m (σ(s), s), for ll (t, s) I [, σ q (b)], where = min { 6 n, 6 m }. EJQTDE, 2009 No. 32, p. 5

6 3 Existence nd Uniqueness In this section, we give the existence nd locl uniqueness of solution of the BVP ()-(2). To prove this result, we define B = E E nd for (y, y 2 ) B, we denote the norm by (y, y 2 ) = y 0 + y 2 0, where E = {y : y C[, σ q (b)]} with the norm y 0 = mx t [,σ q (b)]{ y(t) }, obviously (B,. ) is Bnch spce. Theorem 3. If M stisfies where ɛ = 2mx{ɛ m,ɛ n}, Q Mɛ, ɛ m = mx t [,σ q (b)] nd Q > 0 stisfies Q G m (t, s) s; nd ɛ n = mx G n (t, s) s t [,σ q (b)] mx { f (t, y, y 2 ), f 2 (t, y, y 2 ) }, for t [, σ q (b)], (y,y 2 ) M then the BVP ()-(2) hs solution in the cone P contined in B. Proof: Set P = {(y, y 2 ) B : (y, y 2 ) M} the P is cone in B, Note tht P is closed, bounded nd convex subset of B to which the Schuder fixed point theorem is pplicble. Define T : P B by ( ) T(y, y 2 )(t) := G m (t, s)f (s, y, y 2 ) s, G n (t, s)f 2 (s, y, y 2 ) s := (T m (y, y 2 )(t), T n (y, y 2 )(t)), for t [, σ q (b)]. Obviously the solution of the BVP ()-(2) is the fixed point of opertor T. It cn be shown tht T : P B is continuous. Clim tht T : P P. If (y, y 2 ) P, then T(y, y 2 ) = T m (y, y 2 ) 0 + T n (y, y 2 ) 0 = mx T m(y, y 2 ) + mx T n(y, y 2 ) t [,σ q (b)] t [,σ q (b)] (ɛ m + ɛ n )Q Q ɛ, where Q mx { f (t, y, y 2 ), f 2 (t, y, y 2 ) }, (y,y 2 ) M EJQTDE, 2009 No. 32, p. 6

7 for t [, σ q (b)]. Thus we hve T(y, y 2 ) M, where M stisfies Q Mɛ. Corollry 3.2 If the functions f, f 2, s defined in eqution (), re continuous nd bounded. Then the BVP ()-(2) hs solution. Proof: Choose P > sup{ f (t, y, y 2 ), f 2 (t, y, y 2 ) }, t [, σ q (b)]. Pick M lrge enough so tht P < Mɛ, where ɛ = 2mx{ɛ m,ɛ n}. Then there is number Q > 0 such tht P > Q where Q mx { f (t, y, y 2 ), f 2 (t, y, y 2 ) }, t [, σ q (b)]. (y,y 2 ) M Hence ɛ < M P M Q, nd then the BVP ()-(2) hs solution by Theorem Existence of Multiple Positive Solutions In this section, we estblish the existence of t lest three positive solutions for the system of BVPs ()-(2). And lso we estblish the 2k positive solutions for rbitrry positive integer k. Let B be rel Bnch spce with cone P. A mp S : P [0, ) is sid to be nonnegtive continuous concve functionl on P, if S is continuous nd S(λx + ( λ)y) λs(x) + ( λ)s(y), for ll x, y P nd λ [0, ]. Let nd b be two rel numbers such tht 0 < < b nd S be nonnegtive continuous concve functionl on P. We define the following convex sets P = {y P : y < }, P(S,, b ) = {y P : S(y), y b }. We now stte the fmous Leggett-Willims fixed point theorem. EJQTDE, 2009 No. 32, p. 7

8 Theorem 4. Let T : P c P c be completely continuous nd S be nonnegtive continuous concve functionl on P such tht S(y) y for ll y P c. Suppose tht there exist, b, c, nd d with 0 < d < < b c such tht (i){y P(S,, b ) : S(y) > } nd S(Ty) > for y P(S,, b ), (ii) Ty < d for y d, (iii)s(ty) > for y P(S,, c ) with T(y) > b. Then T hs t lest three fixed points y, y 2, y 3 in P c stisfying y < d, < S(y 2 ), y 3 > d, S(y 3 ) <. For convenience, we let C m = min G m (t, s) s; C n = min s I s I G n (t, s) s. Theorem 4.2 Assume tht there exist rel numbers d 0, d, nd c with 0 < d 0 < d < d < c such tht f (t, y (t), y 2 (t)) < d 0 2ɛ m nd f 2 (t, y (t), y 2 (t)) < d 0 2ɛ n, (8) for t [, σ q (b)] nd (y, y 2 ) [0, d 0 ] [0, d 0 ], f (t, y (t), y 2 (t)) > d 2C m or f 2 (t, y (t), y 2 (t)) > d 2C n, (9) for t I nd (y, y 2 ) [d, d ] [d, d ], f (t, y (t), y 2 (t)) < c 2ɛ m nd f 2 (t, y (t), y 2 (t)) < c 2ɛ n, (0) for t [, σ q (b)] nd (y, y 2 ) [0, c] [0, c]. Then the BVP ()-(2) hs t lest three positive solutions. Proof: We consider the Bnch spce B = E E where E = {y y C[, σ q (b)]} with the norm y 0 = mx t [,σ q (b)] y(t). EJQTDE, 2009 No. 32, p. 8

9 And for (y, y 2 ) B, we denote the norm by (y, y 2 ) = y 0 + y 2 0. Then define cone P in B by P = {(y, y 2 ) B : y (t) 0 nd y 2 (t) 0, t [, σ q (b)]}. For (y, y 2 ) P, we define We denote S(y, y 2 ) = min {y (t)} + min {y 2(t)}. T m (y, y 2 )(t) := T n (y, y 2 )(t) := G m (t, s)f (s, y (s), y 2 (s)) s, G n (t, s)f 2 (s, y (s), y 2 (s)) s, for t [, σ q (b)] nd the opertor T(y, y 2 )(t) := (T m (y, y 2 )(t), T n (y, y 2 )(t)). It is esy to check tht S is nonnegtive continuous concve functionl on P with S(y, y 2 )(t) (y, y 2 ) for (y, y 2 ) P nd tht T : P P is completely continuous nd fixed points of T re solutions of the BVP ()-(2). First, we prove tht if there exists positive number r such tht f (t, y (t), y 2 (t)) < r 2ɛ m nd f 2 (t, y (t), y 2 (t)) < r 2ɛ n for (y, y 2 ) [0, r] [0, r], then T : P r P r. Indeed, if (y, y 2 ) P r, then for t [, σ q (b)]. T(y, y 2 ) = σ ( b) mx G m (t, s)f (s, y (s), y 2 (s)) s t [,σ q (b)] < r 2ɛ m + mx t [,σ q (b)] σ ( b) G m (t, s) s + r 2ɛ n G n (t, s)f 2 (s, y (s), y 2 (s)) s G n (t, s) s = r. Thus, T(y, y 2 ) < r, tht is, T(y, y 2 ) P r. Hence, we hve shown tht if (8) nd (0) hold, then T mps P d0 into P d0 nd P c into P c. Next, we show tht {(y, y 2 ) P(S, d, d ) : S(y, y 2 ) > d } nd S(T(y, y 2 )) > d for ll (y, y 2 ) P(S, d, d ). In fct, the constnt function d + d 2 { (y, y 2 ) P(S, d, d } ) : S(y, y 2 ) > d. EJQTDE, 2009 No. 32, p. 9

10 Moreover, for (y, y 2 ) P(S, d, d ), we hve d (y, y 2 ) y (t) + y 2 (t) min {y (t)} + min {y 2(t)} = S(y, y 2 ) d, for ll t I. Thus, in view of (9) we see tht S(T(y, y 2 )) = min G m (t, s)f (s, y (s), y 2 (s)) s + min G n (t, s)f 2 (s, y (s), y 2 (s)) s { } min G m (t, s)f (s, y (s), y 2 (s)) s s I { } + min G n (t, s)f 2 (s, y (s), y 2 (s)) s s I > d { } min G m (t, s) s + d { min 2C m 2C n s I s I } G n (t, s) s = d, s required. Finlly, we show tht if (y, y 2 ) P(S, d, c) nd T(y, y 2 ) > d, then S(T(y, y 2 )) > d. To see this, we suppose tht (y, y 2 ) P(S, d, c) nd T(y, y 2 ) > d, then, by Lemm 2.3, we hve S(T(y, y 2 )) = min + min + G m (t, s)f (s, y (s), y 2 (s)) s G n (t, s)f 2 (s, y (s), y 2 (s)) s G m (σ(s), s)f (s, y (s), y 2 (s)) s mx t [,σ q (b)] G n (σ(s), s)f 2 (s, y (s), y 2 (s)) s G m (t, s)f (s, y (s), y 2 (s)) s + mx t [,σ q (b)] G m (t, s)f (s, y (s), y 2 (s)) s, EJQTDE, 2009 No. 32, p. 0

11 for ll t [, σ q (b)]. Thus S(T(y, y 2 )) mx t [,σ q (b)] G m (t, s)f (s, y (s), y 2 (s)) s + mx t [,σ q (b)] = T(y, y 2 ) > d = d. G m (t, s)f (s, y (s), y 2 (s)) s To sum up the bove, ll the hypotheses of Theorem 4.2 re stisfied. Hence T hs t lest three fixed points, tht is, the BVP ()-(2) hs t lest three positive solutions (y, y 2 ), (u, u 2 ), nd (w, w 2 ) such tht (y, y 2 ) < d 0, d < min (u, u 2 ), (w, w 2 ) > d 0, min (w, w 2 ) < d. Now, we estblish the existence of t lest 2k positive solutions for the BVP ()-(2), by using induction on k. Theorem 4.3 Let k be n rbitrry positive integer. Assume tht there exist numbers i ( i k) nd b j ( j k ) with 0 < < b < b < 2 < b 2 < b 2 <... < k < b k < b k < k such tht f (t, y (t), y 2 (t)) < i 2ɛ m nd f 2 (t, y (t), y 2 (t)) < i 2ɛ n, () for t [, σ q (b)] nd (y, y 2 ) [0, i ] [0, i ], i k f (t, y (t), y 2 (t)) > b j 2C m or f 2 (t, y (t), y 2 (t)) > b j 2C n (2) for t I nd (y, y 2 ) [b j, b j ] [b j, b j ], j k. Then the BVP ()-(2) hs t lest 2k positive solutions in P k. Proof: We use induction on k. First, for k =, we know from () tht T : P P, then, it follows from Schuder fixed point theorem tht the BVP ()-(2) hs t lest one positive solution in P. Next, we ssume tht this conclusion holds for k = r. In order to prove tht this conclusion holds for k = r +, we suppose tht there exist numbers i ( i r + ) nd EJQTDE, 2009 No. 32, p.

12 b j ( j r) with 0 < < b < b < 2 < b 2 < b 2 <... < r < b r < br < r+ such tht f (t, y (t), y 2 (t)) < i 2ɛ m nd f 2 (t, y (t), y 2 (t)) < i 2ɛ n, (3) for t [, σ q (b)] nd (y, y 2 ) [0, i ] [0, i ], i r + f (t, y (t), y 2 (t)) > b j 2C m or f 2 (t, y (t), y 2 (t)) > b j 2C n (4) for t I nd (y, y 2 ) [b j, b j ] [b j, b j ], j r. By ssumption, the BVP ()-(2) hs t lest 2r positive solutions (u i, u i )(i =, 2,..., 2r ) in P r. At the sme time, it follows from Theorem 4.2, (3) nd (4) tht the BVP ()-(2) hs t lest three positive solutions (u, u ), (v, v 2 ) nd (w, w 2 ) in P r+ such tht, (u, u ) < r, b r < min (v (t), v 2 (t)), (w, w 2 ) > r, min (w (t), w 2 (t)) < b r. Obviously, (v, v 2 ) nd (w, w 2 ) re different from (u i, u i )(i =, 2,..., 2r ). Therefore, the BVP()-(2) hs t lest 2r + positive solutions in P r+ which shows tht this conclusion lso holds for k = r +. Acknowledgement One of the uthors (P.Murli) is thnkful to CSIR, Indi for wrding SRF. The uthors thnk the referees for their vluble suggestions. References [] R. P. Agrwl, D. O Regn, nd P. J. Y. Wong, Positive Solutions of Differentil, Difference nd Integrl Equtions, Kluwer Acdemic Publishers, Dordrecht, The Netherlnds, 999. [2] D. R. Anderson nd R. I. Avery, Multiple positive solutions to thirdorder discrete focl boundry vlue problem, J. Computers nd Mthemtics with Applictions, 42(200), [3] R. I. Avery nd A. C. Peterson, Multiple positive solutions of discrete second order conjugte problem, Pnmer. Mth. J., 8(998), -2. [4] M. Bohner nd A. C. Peterson, Dynmic Equtions on Time scles, An Introduction with Applictions, Birkhuser, Boston, MA, (200). EJQTDE, 2009 No. 32, p. 2

13 [5] P. W. Eloe nd J. Henderson, Positive solutions for (n-,) conjugte boundry vlue problems, Nonliner Anl., 28(997), [6] P. W. Eloe nd J. Henderson, Positive solutions nd nonliner (k,n-k) conjugte eigenvlue problems, J. Diff. Eqn. Dyn. Syst., 6(998), [7] L. H. Erbe nd H. Wng, On the existence of positive solutions of ordinry differentil equtions, Proc. Amer. Mth. Soc., 20(994), [8] K. M. Fick nd J. Henderson, Existence of positive solutions of 2nth order eigenvlue problem, Nonliner Diff. Eqn. Theory-Methods nd Applictions, 7(2002), no. & 2, [9] B. Hopkins nd N. Kosmtov, Third order boundry vlue problem with sign-chnging solution, Nonliner Anlysis, 67(2007), [0] S. Li, Positive solutions of nonliner singulr third order two-point boundry vlue problem, J. Mth. Anl. Appl., 323(2006), [] F. Merdivenci Atici nd G. Sh. Guseinov, Positive periodic solutions for nonliner differnce equlions with periodic coefficients, J. Mth. Anl. Appl., 232(999), [2] A. C. Peterson, Y. N. Rffoul, nd C. C. Tisdell, Three point boundry vlue problems on time scles, J.Diff. Eqn. Appl., 0(2004), [3] K. R. Prsd nd P. Murli, Eigenvlue intervls for n th order differentil equtions on time scles, Inter. J. Pure nd Appl. Mth., 44(2008), no. 5, [4] L. Snchez, Positive solutions for clss of semiliner two-point boundry vlue problems, Bull. Austrl. Mth. Soc., 45(992), [5] H. R. Sun nd W. T. Li, Positive solutions for nonliner three-point boundry vlue problems on time scles, J. Mth. Anl. Appl., 299(2004), (Received Mrch 4, 2009) EJQTDE, 2009 No. 32, p. 3